src/HOL/Divides.thy
author haftmann
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using only an relation predicate to construct div and mod
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(*  Title:      HOL/Divides.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1999  University of Cambridge
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*)
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header {* The division operators div,  mod and the divides relation dvd *}
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theory Divides
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imports Nat Power Product_Type Wellfounded_Recursion
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uses "~~/src/Provers/Arith/cancel_div_mod.ML"
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begin
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subsection {* Syntactic division operations *}
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class div = times +
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  fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
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  fixes mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)
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begin
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definition
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  dvd  :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "dvd" 50)
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where
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  [code func del]: "m dvd n \<longleftrightarrow> (\<exists>k. n = m * k)"
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end
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subsection {* Abstract divisibility in commutative semirings. *}
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class semiring_div = comm_semiring_1_cancel + div + 
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  assumes mod_div_equality: "a div b * b + a mod b = a"
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    and div_by_0: "a div 0 = 0"
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    and mult_div: "b \<noteq> 0 \<Longrightarrow> a * b div b = a"
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begin
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text {* @{const div} and @{const mod} *}
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lemma div_by_1: "a div 1 = a"
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  using mult_div [of 1 a] zero_neq_one by simp
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lemma mod_by_1: "a mod 1 = 0"
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proof -
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  from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp
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  then have "a + a mod 1 = a + 0" by simp
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  then show ?thesis by (rule add_left_imp_eq)
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qed
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lemma mod_by_0: "a mod 0 = a"
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  using mod_div_equality [of a zero] by simp
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lemma mult_mod: "a * b mod b = 0"
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proof (cases "b = 0")
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  case True then show ?thesis by (simp add: mod_by_0)
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next
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  case False with mult_div have abb: "a * b div b = a" .
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  from mod_div_equality have "a * b div b * b + a * b mod b = a * b" .
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  with abb have "a * b + a * b mod b = a * b + 0" by simp
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  then show ?thesis by (rule add_left_imp_eq)
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qed
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lemma mod_self: "a mod a = 0"
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  using mult_mod [of one] by simp
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lemma div_self: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
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  using mult_div [of _ one] by simp
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lemma div_0: "0 div a = 0"
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proof (cases "a = 0")
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  case True then show ?thesis by (simp add: div_by_0)
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next
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  case False with mult_div have "0 * a div a = 0" .
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  then show ?thesis by simp
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qed
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lemma mod_0: "0 mod a = 0"
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  using mod_div_equality [of zero a] div_0 by simp 
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lemma mod_div_equality2: "b * (a div b) + a mod b = a"
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  unfolding mult_commute [of b]
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  by (rule mod_div_equality)
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lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
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  by (simp add: mod_div_equality)
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lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"
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  by (simp add: mod_div_equality2)
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text {* The @{const dvd} relation *}
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lemma dvdI [intro?]: "a = b * c \<Longrightarrow> b dvd a"
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  unfolding dvd_def ..
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lemma dvdE [elim?]: "b dvd a \<Longrightarrow> (\<And>c. a = b * c \<Longrightarrow> P) \<Longrightarrow> P"
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  unfolding dvd_def by blast 
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lemma dvd_def_mod [code func]: "a dvd b \<longleftrightarrow> b mod a = 0"
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proof
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  assume "b mod a = 0"
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  with mod_div_equality [of b a] have "b div a * a = b" by simp
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  then have "b = a * (b div a)" unfolding mult_commute ..
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  then have "\<exists>c. b = a * c" ..
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  then show "a dvd b" unfolding dvd_def .
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next
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  assume "a dvd b"
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  then have "\<exists>c. b = a * c" unfolding dvd_def .
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  then obtain c where "b = a * c" ..
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  then have "b mod a = a * c mod a" by simp
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  then have "b mod a = c * a mod a" by (simp add: mult_commute)
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  then show "b mod a = 0" by (simp add: mult_mod)
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qed
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lemma dvd_refl: "a dvd a"
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  unfolding dvd_def_mod mod_self ..
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lemma dvd_trans:
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  assumes "a dvd b" and "b dvd c"
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  shows "a dvd c"
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proof -
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  from assms obtain v where "b = a * v" unfolding dvd_def by auto
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  moreover from assms obtain w where "c = b * w" unfolding dvd_def by auto
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  ultimately have "c = a * (v * w)" by (simp add: mult_assoc)
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  then show ?thesis unfolding dvd_def ..
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qed
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lemma zero_dvd_iff [noatp]: "0 dvd a \<longleftrightarrow> a = 0"
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  unfolding dvd_def by simp
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lemma dvd_0: "a dvd 0"
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unfolding dvd_def proof
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  show "0 = a * 0" by simp
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qed
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lemma one_dvd: "1 dvd a"
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  unfolding dvd_def by simp
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lemma dvd_mult: "a dvd c \<Longrightarrow> a dvd (b * c)"
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  unfolding dvd_def by (blast intro: mult_left_commute)
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lemma dvd_mult2: "a dvd b \<Longrightarrow> a dvd (b * c)"
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  apply (subst mult_commute)
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  apply (erule dvd_mult)
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  done
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lemma dvd_triv_right: "a dvd b * a"
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  by (rule dvd_mult) (rule dvd_refl)
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lemma dvd_triv_left: "a dvd a * b"
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  by (rule dvd_mult2) (rule dvd_refl)
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lemma mult_dvd_mono: "a dvd c \<Longrightarrow> b dvd d \<Longrightarrow> a * b dvd c * d"
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  apply (unfold dvd_def, clarify)
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  apply (rule_tac x = "k * ka" in exI)
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  apply (simp add: mult_ac)
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  done
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lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c"
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  by (simp add: dvd_def mult_assoc, blast)
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lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c"
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  unfolding mult_ac [of a] by (rule dvd_mult_left)
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end
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subsection {* Division on @{typ nat} *}
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text {*
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  We define @{const div} and @{const mod} on @{typ nat} by means
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  of a characteristic relation with two input arguments
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  @{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments
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  @{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder).
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*}
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definition divmod_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where
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  "divmod_rel m n q r \<longleftrightarrow> m = q * n + r \<and> (if n > 0 then 0 \<le> r \<and> r < n else q = 0)"
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text {* @{const divmod_rel} is total: *}
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lemma divmod_rel_ex:
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  obtains q r where "divmod_rel m n q r"
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proof (cases "n = 0")
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  case True with that show thesis
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    by (auto simp add: divmod_rel_def)
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next
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  case False
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  have "\<exists>q r. m = q * n + r \<and> r < n"
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  proof (induct m)
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    case 0 with `n \<noteq> 0`
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    have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp
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    then show ?case by blast
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  next
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    case (Suc m) then obtain q' r'
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      where m: "m = q' * n + r'" and n: "r' < n" by auto
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    then show ?case proof (cases "Suc r' < n")
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      case True
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      from m n have "Suc m = q' * n + Suc r'" by simp
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      with True show ?thesis by blast
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    next
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      case False then have "n \<le> Suc r'" by auto
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      moreover from n have "Suc r' \<le> n" by auto
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      ultimately have "n = Suc r'" by auto
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      with m have "Suc m = Suc q' * n + 0" by simp
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      with `n \<noteq> 0` show ?thesis by blast
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    qed
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  qed
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  with that show thesis
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    using `n \<noteq> 0` by (auto simp add: divmod_rel_def)
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qed
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text {* @{const divmod_rel} is injective: *}
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lemma divmod_rel_unique_div:
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  assumes "divmod_rel m n q r"
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    and "divmod_rel m n q' r'"
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  shows "q = q'"
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proof (cases "n = 0")
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  case True with assms show ?thesis
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    by (simp add: divmod_rel_def)
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next
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  case False
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  have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)"
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  apply (rule leI)
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  apply (subst less_iff_Suc_add)
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  apply (auto simp add: add_mult_distrib)
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  done
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  from `n \<noteq> 0` assms show ?thesis
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    by (auto simp add: divmod_rel_def
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      intro: order_antisym dest: aux sym)
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qed
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lemma divmod_rel_unique_mod:
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  assumes "divmod_rel m n q r"
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    and "divmod_rel m n q' r'"
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  shows "r = r'"
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proof -
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  from assms have "q = q'" by (rule divmod_rel_unique_div)
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  with assms show ?thesis by (simp add: divmod_rel_def)
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qed
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text {*
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  We instantiate divisibility on the natural numbers by
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  means of @{const divmod_rel}:
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*}
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a52309ac4a4d added class semiring_div
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instantiation nat :: semiring_div
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begin
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definition divmod :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
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  [code func del]: "divmod m n = (THE (q, r). divmod_rel m n q r)"
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definition div_nat where
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  "m div n = fst (divmod m n)"
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definition mod_nat where
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  "m mod n = snd (divmod m n)"
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lemma divmod_div_mod:
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  "divmod m n = (m div n, m mod n)"
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  unfolding div_nat_def mod_nat_def by simp
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lemma divmod_eq:
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  assumes "divmod_rel m n q r" 
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  shows "divmod m n = (q, r)"
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  using assms by (auto simp add: divmod_def
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    dest: divmod_rel_unique_div divmod_rel_unique_mod)
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lemma div_eq:
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  assumes "divmod_rel m n q r" 
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  shows "m div n = q"
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  using assms by (auto dest: divmod_eq simp add: div_nat_def)
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lemma mod_eq:
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  assumes "divmod_rel m n q r" 
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   274
  shows "m mod n = r"
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  using assms by (auto dest: divmod_eq simp add: mod_nat_def)
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lemma divmod_rel: "divmod_rel m n (m div n) (m mod n)"
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proof -
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   279
  from divmod_rel_ex
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    obtain q r where rel: "divmod_rel m n q r" .
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  moreover with div_eq mod_eq have "m div n = q" and "m mod n = r"
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   282
    by simp_all
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  ultimately show ?thesis by simp
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qed
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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   285
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lemma divmod_zero:
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  "divmod m 0 = (0, m)"
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   288
proof -
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   289
  from divmod_rel [of m 0] show ?thesis
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   290
    unfolding divmod_div_mod divmod_rel_def by simp
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qed
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   292
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lemma divmod_base:
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   294
  assumes "m < n"
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   295
  shows "divmod m n = (0, m)"
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   296
proof -
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   297
  from divmod_rel [of m n] show ?thesis
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   298
    unfolding divmod_div_mod divmod_rel_def
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   299
    using assms by (cases "m div n = 0")
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      (auto simp add: gr0_conv_Suc [of "m div n"])
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qed
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   302
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lemma divmod_step:
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   304
  assumes "0 < n" and "n \<le> m"
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   305
  shows "divmod m n = (Suc ((m - n) div n), (m - n) mod n)"
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   306
proof -
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   307
  from divmod_rel have divmod_m_n: "divmod_rel m n (m div n) (m mod n)" .
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   308
  with assms have m_div_n: "m div n \<ge> 1"
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   309
    by (cases "m div n") (auto simp add: divmod_rel_def)
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   310
  from assms divmod_m_n have "divmod_rel (m - n) n (m div n - 1) (m mod n)"
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   311
    by (cases "m div n") (auto simp add: divmod_rel_def)
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   312
  with divmod_eq have "divmod (m - n) n = (m div n - 1, m mod n)" by simp
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   313
  moreover from divmod_div_mod have "divmod (m - n) n = ((m - n) div n, (m - n) mod n)" .
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   314
  ultimately have "m div n = Suc ((m - n) div n)"
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   315
    and "m mod n = (m - n) mod n" using m_div_n by simp_all
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   316
  then show ?thesis using divmod_div_mod by simp
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   317
qed
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   318
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   319
text {* The ''recursion┬┤┬┤ equations for @{const div} and @{const mod} *}
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   320
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   321
lemma div_less [simp]:
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   322
  fixes m n :: nat
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  assumes "m < n"
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  shows "m div n = 0"
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  using assms divmod_base divmod_div_mod by simp
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lemma le_div_geq:
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  fixes m n :: nat
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  assumes "0 < n" and "n \<le> m"
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  shows "m div n = Suc ((m - n) div n)"
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  using assms divmod_step divmod_div_mod by simp
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lemma mod_less [simp]:
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  fixes m n :: nat
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  assumes "m < n"
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  shows "m mod n = m"
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  using assms divmod_base divmod_div_mod by simp
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lemma le_mod_geq:
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  fixes m n :: nat
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  assumes "n \<le> m"
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  shows "m mod n = (m - n) mod n"
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  using assms divmod_step divmod_div_mod by (cases "n = 0") simp_all
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instance proof
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  fix m n :: nat show "m div n * n + m mod n = m"
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    using divmod_rel [of m n] by (simp add: divmod_rel_def)
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next
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   349
  fix n :: nat show "n div 0 = 0"
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   350
    using divmod_zero divmod_div_mod [of n 0] by simp
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next
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  fix m n :: nat assume "n \<noteq> 0" then show "m * n div n = m"
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    by (induct m) (simp_all add: le_div_geq)
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qed
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   356
end
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text {* Simproc for cancelling @{const div} and @{const mod} *}
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   359
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lemmas mod_div_equality = semiring_div_class.times_div_mod_plus_zero_one.mod_div_equality [of "m\<Colon>nat" n, standard]
26062
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lemmas mod_div_equality2 = mod_div_equality2 [of "n\<Colon>nat" m, standard]
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   362
lemmas div_mod_equality = div_mod_equality [of "m\<Colon>nat" n k, standard]
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   363
lemmas div_mod_equality2 = div_mod_equality2 [of "m\<Colon>nat" n k, standard]
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   364
a52309ac4a4d added class semiring_div
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   365
ML {*
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   366
structure CancelDivModData =
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   367
struct
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   368
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val div_name = @{const_name div};
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   370
val mod_name = @{const_name mod};
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val mk_binop = HOLogic.mk_binop;
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val mk_sum = ArithData.mk_sum;
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   373
val dest_sum = ArithData.dest_sum;
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   374
a52309ac4a4d added class semiring_div
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   375
(*logic*)
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   377
val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}]
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   378
a52309ac4a4d added class semiring_div
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   379
val trans = trans
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   380
a52309ac4a4d added class semiring_div
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   381
val prove_eq_sums =
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   382
  let val simps = @{thm add_0} :: @{thm add_0_right} :: @{thms add_ac}
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diff changeset
   383
  in ArithData.prove_conv all_tac (ArithData.simp_all_tac simps) end;
25942
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   384
a52309ac4a4d added class semiring_div
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   385
end;
a52309ac4a4d added class semiring_div
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   386
a52309ac4a4d added class semiring_div
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   387
structure CancelDivMod = CancelDivModFun(CancelDivModData);
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diff changeset
   388
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   389
val cancel_div_mod_proc = Simplifier.simproc @{theory}
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haftmann
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   390
  "cancel_div_mod" ["(m::nat) + n"] (K CancelDivMod.proc);
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diff changeset
   391
a52309ac4a4d added class semiring_div
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   392
Addsimprocs[cancel_div_mod_proc];
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   393
*}
a52309ac4a4d added class semiring_div
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   394
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   395
text {* code generator setup *}
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diff changeset
   396
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   397
lemma divmod_if [code]: "divmod m n = (if n = 0 \<or> m < n then (0, m) else
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haftmann
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diff changeset
   398
  let (q, r) = divmod (m - n) n in (Suc q, r))"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   399
  by (simp add: divmod_zero divmod_base divmod_step)
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diff changeset
   400
    (simp add: divmod_div_mod)
fbc60cd02ae2 using only an relation predicate to construct div and mod
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diff changeset
   401
fbc60cd02ae2 using only an relation predicate to construct div and mod
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diff changeset
   402
code_modulename SML
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haftmann
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diff changeset
   403
  Divides Nat
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diff changeset
   404
fbc60cd02ae2 using only an relation predicate to construct div and mod
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diff changeset
   405
code_modulename OCaml
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diff changeset
   406
  Divides Nat
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parents: 26072
diff changeset
   407
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
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diff changeset
   408
code_modulename Haskell
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haftmann
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diff changeset
   409
  Divides Nat
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diff changeset
   410
fbc60cd02ae2 using only an relation predicate to construct div and mod
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diff changeset
   411
fbc60cd02ae2 using only an relation predicate to construct div and mod
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diff changeset
   412
subsubsection {* Quotient *}
fbc60cd02ae2 using only an relation predicate to construct div and mod
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diff changeset
   413
fbc60cd02ae2 using only an relation predicate to construct div and mod
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   414
lemmas DIVISION_BY_ZERO_DIV [simp] = div_by_0 [of "a\<Colon>nat", standard]
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diff changeset
   415
lemmas div_0 [simp] = semiring_div_class.div_0 [of "n\<Colon>nat", standard]
fbc60cd02ae2 using only an relation predicate to construct div and mod
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diff changeset
   416
fbc60cd02ae2 using only an relation predicate to construct div and mod
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diff changeset
   417
lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
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diff changeset
   418
  by (simp add: le_div_geq linorder_not_less)
fbc60cd02ae2 using only an relation predicate to construct div and mod
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diff changeset
   419
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
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diff changeset
   420
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
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diff changeset
   421
  by (simp add: div_geq)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
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diff changeset
   422
fbc60cd02ae2 using only an relation predicate to construct div and mod
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diff changeset
   423
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
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diff changeset
   424
  by (rule mult_div) simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   425
fbc60cd02ae2 using only an relation predicate to construct div and mod
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diff changeset
   426
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   427
  by (simp add: mult_commute)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
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diff changeset
   428
25942
a52309ac4a4d added class semiring_div
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diff changeset
   429
a52309ac4a4d added class semiring_div
haftmann
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diff changeset
   430
subsubsection {* Remainder *}
a52309ac4a4d added class semiring_div
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diff changeset
   431
a52309ac4a4d added class semiring_div
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diff changeset
   432
lemmas DIVISION_BY_ZERO_MOD [simp] = mod_by_0 [of "a\<Colon>nat", standard]
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   433
lemmas mod_0 [simp] = semiring_div_class.mod_0 [of "n\<Colon>nat", standard]
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   434
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
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diff changeset
   435
lemma mod_less_divisor [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   436
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   437
  assumes "n > 0"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   438
  shows "m mod n < (n::nat)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   439
  using assms divmod_rel unfolding divmod_rel_def by auto
14267
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paulson
parents: 14208
diff changeset
   440
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   441
lemma mod_less_eq_dividend [simp]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   442
  fixes m n :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   443
  shows "m mod n \<le> m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   444
proof (rule add_leD2)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   445
  from mod_div_equality have "m div n * n + m mod n = m" .
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   446
  then show "m div n * n + m mod n \<le> m" by auto
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   447
qed
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
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diff changeset
   448
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
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diff changeset
   449
lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m - n) mod n"
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   450
  by (simp add: le_mod_geq linorder_not_less)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   451
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
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diff changeset
   452
lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m - n) mod n)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   453
  by (simp add: le_mod_geq)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   454
14267
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paulson
parents: 14208
diff changeset
   455
lemma mod_1 [simp]: "m mod Suc 0 = 0"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   456
  by (induct m) (simp_all add: mod_geq)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   457
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   458
lemmas mod_self [simp] = semiring_div_class.mod_self [of "n\<Colon>nat", standard]
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paulson
parents: 14208
diff changeset
   459
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   460
lemma mod_add_self2 [simp]: "(m+n) mod n = m mod (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   461
  apply (subgoal_tac "(n + m) mod n = (n+m-n) mod n")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   462
   apply (simp add: add_commute)
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   463
  apply (subst le_mod_geq [symmetric], simp_all)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   464
  done
14267
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paulson
parents: 14208
diff changeset
   465
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   466
lemma mod_add_self1 [simp]: "(n+m) mod n = m mod (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   467
  by (simp add: add_commute mod_add_self2)
14267
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paulson
parents: 14208
diff changeset
   468
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   469
lemma mod_mult_self1 [simp]: "(m + k*n) mod n = m mod (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   470
  by (induct k) (simp_all add: add_left_commute [of _ n])
14267
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paulson
parents: 14208
diff changeset
   471
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   472
lemma mod_mult_self2 [simp]: "(m + n*k) mod n = m mod (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   473
  by (simp add: mult_commute mod_mult_self1)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   474
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   475
lemma mod_mult_distrib: "(m mod n) * (k\<Colon>nat) = (m * k) mod (n * k)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   476
  apply (cases "n = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   477
  apply (cases "k = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   478
  apply (induct m rule: nat_less_induct)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   479
  apply (subst mod_if, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   480
  apply (simp add: mod_geq diff_mult_distrib)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   481
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   482
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   483
lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   484
  by (simp add: mult_commute [of k] mod_mult_distrib)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   485
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   486
lemma mod_mult_self_is_0 [simp]: "(m*n) mod n = (0::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   487
  apply (cases "n = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   488
  apply (induct m, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   489
  apply (rename_tac k)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   490
  apply (cut_tac m = "k * n" and n = n in mod_add_self2)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   491
  apply (simp add: add_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   492
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   493
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   494
lemma mod_mult_self1_is_0 [simp]: "(n*m) mod n = (0::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   495
  by (simp add: mult_commute mod_mult_self_is_0)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   496
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   497
(* a simple rearrangement of mod_div_equality: *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   498
lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   499
  by (cut_tac m = m and n = n in mod_div_equality2, arith)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   500
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   501
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   502
  apply (drule mod_less_divisor [where m = m])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   503
  apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   504
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   505
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   506
subsubsection {* Quotient and Remainder *}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   507
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   508
lemma mod_div_decomp:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   509
  fixes n k :: nat
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   510
  obtains m q where "m = n div k" and "q = n mod k"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   511
    and "n = m * k + q"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   512
proof -
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   513
  from mod_div_equality have "n = n div k * k + n mod k" by auto
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   514
  moreover have "n div k = n div k" ..
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   515
  moreover have "n mod k = n mod k" ..
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   516
  note that ultimately show thesis by blast
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   517
qed
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   518
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   519
lemma divmod_rel_mult1_eq:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   520
  "[| divmod_rel b c q r; c > 0 |]
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   521
   ==> divmod_rel (a*b) c (a*q + a*r div c) (a*r mod c)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   522
by (auto simp add: split_ifs mult_ac divmod_rel_def add_mult_distrib2)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   523
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   524
lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   525
apply (cases "c = 0", simp)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   526
apply (blast intro: divmod_rel [THEN divmod_rel_mult1_eq, THEN div_eq])
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   527
done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   528
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   529
lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   530
apply (cases "c = 0", simp)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   531
apply (blast intro: divmod_rel [THEN divmod_rel_mult1_eq, THEN mod_eq])
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   532
done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   533
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   534
lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   535
  apply (rule trans)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   536
   apply (rule_tac s = "b*a mod c" in trans)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   537
    apply (rule_tac [2] mod_mult1_eq)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   538
   apply (simp_all add: mult_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   539
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   540
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   541
lemma mod_mult_distrib_mod:
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   542
  "(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c"
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   543
apply (rule mod_mult1_eq' [THEN trans])
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   544
apply (rule mod_mult1_eq)
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   545
done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   546
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   547
lemma divmod_rel_add1_eq:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   548
  "[| divmod_rel a c aq ar; divmod_rel b c bq br;  c > 0 |]
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   549
   ==> divmod_rel (a + b) c (aq + bq + (ar+br) div c) ((ar + br) mod c)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   550
by (auto simp add: split_ifs mult_ac divmod_rel_def add_mult_distrib2)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   551
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   552
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   553
lemma div_add1_eq:
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   554
  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   555
apply (cases "c = 0", simp)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   556
apply (blast intro: divmod_rel_add1_eq [THEN div_eq] divmod_rel)
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   557
done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   558
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   559
lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   560
apply (cases "c = 0", simp)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   561
apply (blast intro: divmod_rel_add1_eq [THEN mod_eq] divmod_rel)
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   562
done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   563
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   564
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   565
  apply (cut_tac m = q and n = c in mod_less_divisor)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   566
  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   567
  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   568
  apply (simp add: add_mult_distrib2)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   569
  done
10559
d3fd54fc659b many new div and mod properties (borrowed from Integ/IntDiv)
paulson
parents: 10214
diff changeset
   570
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   571
lemma divmod_rel_mult2_eq: "[| divmod_rel a b q r;  0 < b;  0 < c |]
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   572
      ==> divmod_rel a (b*c) (q div c) (b*(q mod c) + r)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   573
  by (auto simp add: mult_ac divmod_rel_def add_mult_distrib2 [symmetric] mod_lemma)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   574
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   575
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   576
  apply (cases "b = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   577
  apply (cases "c = 0", simp)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   578
  apply (force simp add: divmod_rel [THEN divmod_rel_mult2_eq, THEN div_eq])
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   579
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   580
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   581
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   582
  apply (cases "b = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   583
  apply (cases "c = 0", simp)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   584
  apply (auto simp add: mult_commute divmod_rel [THEN divmod_rel_mult2_eq, THEN mod_eq])
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   585
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   586
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   587
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   588
subsubsection{*Cancellation of Common Factors in Division*}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   589
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   590
lemma div_mult_mult_lemma:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   591
    "[| (0::nat) < b;  0 < c |] ==> (c*a) div (c*b) = a div b"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   592
  by (auto simp add: div_mult2_eq)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   593
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   594
lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   595
  apply (cases "b = 0")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   596
  apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   597
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   598
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   599
lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   600
  apply (drule div_mult_mult1)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   601
  apply (auto simp add: mult_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   602
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   603
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   604
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   605
subsubsection{*Further Facts about Quotient and Remainder*}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   606
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   607
lemma div_1 [simp]: "m div Suc 0 = m"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   608
  by (induct m) (simp_all add: div_geq)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   609
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   610
lemmas div_self [simp] = semiring_div_class.div_self [of "n\<Colon>nat", standard]
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   611
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   612
lemma div_add_self2: "0<n ==> (m+n) div n = Suc (m div n)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   613
  apply (subgoal_tac "(n + m) div n = Suc ((n+m-n) div n) ")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   614
   apply (simp add: add_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   615
  apply (subst div_geq [symmetric], simp_all)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   616
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   617
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   618
lemma div_add_self1: "0<n ==> (n+m) div n = Suc (m div n)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   619
  by (simp add: add_commute div_add_self2)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   620
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   621
lemma div_mult_self1 [simp]: "!!n::nat. 0<n ==> (m + k*n) div n = k + m div n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   622
  apply (subst div_add1_eq)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   623
  apply (subst div_mult1_eq, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   624
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   625
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   626
lemma div_mult_self2 [simp]: "0<n ==> (m + n*k) div n = k + m div (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   627
  by (simp add: mult_commute div_mult_self1)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   628
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   629
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   630
(* Monotonicity of div in first argument *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   631
lemma div_le_mono [rule_format (no_asm)]:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   632
    "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   633
apply (case_tac "k=0", simp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   634
apply (induct "n" rule: nat_less_induct, clarify)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   635
apply (case_tac "n<k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   636
(* 1  case n<k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   637
apply simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   638
(* 2  case n >= k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   639
apply (case_tac "m<k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   640
(* 2.1  case m<k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   641
apply simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   642
(* 2.2  case m>=k *)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   643
apply (simp add: div_geq diff_le_mono)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   644
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   645
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   646
(* Antimonotonicity of div in second argument *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   647
lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   648
apply (subgoal_tac "0<n")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   649
 prefer 2 apply simp
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   650
apply (induct_tac k rule: nat_less_induct)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   651
apply (rename_tac "k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   652
apply (case_tac "k<n", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   653
apply (subgoal_tac "~ (k<m) ")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   654
 prefer 2 apply simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   655
apply (simp add: div_geq)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   656
apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   657
 prefer 2
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   658
 apply (blast intro: div_le_mono diff_le_mono2)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   659
apply (rule le_trans, simp)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   660
apply (simp)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   661
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   662
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   663
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   664
apply (case_tac "n=0", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   665
apply (subgoal_tac "m div n \<le> m div 1", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   666
apply (rule div_le_mono2)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   667
apply (simp_all (no_asm_simp))
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   668
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   669
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   670
(* Similar for "less than" *)
17085
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   671
lemma div_less_dividend [rule_format]:
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   672
     "!!n::nat. 1<n ==> 0 < m --> m div n < m"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   673
apply (induct_tac m rule: nat_less_induct)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   674
apply (rename_tac "m")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   675
apply (case_tac "m<n", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   676
apply (subgoal_tac "0<n")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   677
 prefer 2 apply simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   678
apply (simp add: div_geq)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   679
apply (case_tac "n<m")
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   680
 apply (subgoal_tac "(m-n) div n < (m-n) ")
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   681
  apply (rule impI less_trans_Suc)+
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   682
apply assumption
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   683
  apply (simp_all)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   684
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   685
17085
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   686
declare div_less_dividend [simp]
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   687
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   688
text{*A fact for the mutilated chess board*}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   689
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   690
apply (case_tac "n=0", simp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   691
apply (induct "m" rule: nat_less_induct)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   692
apply (case_tac "Suc (na) <n")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   693
(* case Suc(na) < n *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   694
apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   695
(* case n \<le> Suc(na) *)
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16733
diff changeset
   696
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   697
apply (auto simp add: Suc_diff_le le_mod_geq)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   698
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   699
14437
92f6aa05b7bb some new results
paulson
parents: 14430
diff changeset
   700
lemma nat_mod_div_trivial [simp]: "m mod n div n = (0 :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   701
  by (cases "n = 0") auto
14437
92f6aa05b7bb some new results
paulson
parents: 14430
diff changeset
   702
92f6aa05b7bb some new results
paulson
parents: 14430
diff changeset
   703
lemma nat_mod_mod_trivial [simp]: "m mod n mod n = (m mod n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   704
  by (cases "n = 0") auto
14437
92f6aa05b7bb some new results
paulson
parents: 14430
diff changeset
   705
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   706
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   707
(* Antimonotonicity of div in second argument *)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   708
lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   709
apply (subgoal_tac "0<n")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   710
 prefer 2 apply simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   711
apply (induct_tac k rule: nat_less_induct)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   712
apply (rename_tac "k")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   713
apply (case_tac "k<n", simp)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   714
apply (subgoal_tac "~ (k<m) ")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   715
 prefer 2 apply simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   716
apply (simp add: div_geq)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   717
apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   718
 prefer 2
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   719
 apply (blast intro: div_le_mono diff_le_mono2)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   720
apply (rule le_trans, simp)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   721
apply (simp)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   722
done
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   723
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   724
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   725
apply (case_tac "n=0", simp)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   726
apply (subgoal_tac "m div n \<le> m div 1", simp)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   727
apply (rule div_le_mono2)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   728
apply (simp_all (no_asm_simp))
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   729
done
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   730
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   731
(* Similar for "less than" *)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   732
lemma div_less_dividend [rule_format]:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   733
     "!!n::nat. 1<n ==> 0 < m --> m div n < m"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   734
apply (induct_tac m rule: nat_less_induct)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   735
apply (rename_tac "m")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   736
apply (case_tac "m<n", simp)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   737
apply (subgoal_tac "0<n")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   738
 prefer 2 apply simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   739
apply (simp add: div_geq)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   740
apply (case_tac "n<m")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   741
 apply (subgoal_tac "(m-n) div n < (m-n) ")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   742
  apply (rule impI less_trans_Suc)+
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   743
apply assumption
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   744
  apply (simp_all)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   745
done
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   746
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   747
declare div_less_dividend [simp]
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   748
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   749
text{*A fact for the mutilated chess board*}
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   750
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   751
apply (case_tac "n=0", simp)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   752
apply (induct "m" rule: nat_less_induct)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   753
apply (case_tac "Suc (na) <n")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   754
(* case Suc(na) < n *)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   755
apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   756
(* case n \<le> Suc(na) *)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   757
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   758
apply (auto simp add: Suc_diff_le le_mod_geq)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   759
done
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   760
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   761
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   762
subsubsection{*The Divides Relation*}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   763
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   764
lemma dvdI [intro?]: "n = m * k ==> m dvd n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   765
  unfolding dvd_def by blast
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   766
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   767
lemma dvdE [elim?]: "!!P. [|m dvd n;  !!k. n = m*k ==> P|] ==> P"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   768
  unfolding dvd_def by blast
13152
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
   769
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   770
lemma dvd_0_right [iff]: "m dvd (0::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   771
  unfolding dvd_def by (blast intro: mult_0_right [symmetric])
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   772
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   773
lemma dvd_0_left: "0 dvd m ==> m = (0::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   774
  by (force simp add: dvd_def)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   775
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   776
lemma dvd_0_left_iff [iff]: "(0 dvd (m::nat)) = (m = 0)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   777
  by (blast intro: dvd_0_left)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   778
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24268
diff changeset
   779
declare dvd_0_left_iff [noatp]
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24268
diff changeset
   780
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   781
lemma dvd_1_left [iff]: "Suc 0 dvd k"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   782
  unfolding dvd_def by simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   783
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   784
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   785
  by (simp add: dvd_def)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   786
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   787
lemmas dvd_refl [simp] = semiring_div_class.dvd_refl [of "m\<Colon>nat", standard]
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   788
lemmas dvd_trans [trans] = semiring_div_class.dvd_trans [of "m\<Colon>nat" n p, standard]
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   789
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   790
lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   791
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   792
  by (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   793
23684
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
   794
text {* @{term "op dvd"} is a partial order *}
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
   795
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   796
interpretation dvd: order ["op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> n \<noteq> m"]
23684
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
   797
  by unfold_locales (auto intro: dvd_trans dvd_anti_sym)
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
   798
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   799
lemma dvd_add: "[| k dvd m; k dvd n |] ==> k dvd (m+n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   800
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   801
  by (blast intro: add_mult_distrib2 [symmetric])
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   802
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   803
lemma dvd_diff: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   804
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   805
  by (blast intro: diff_mult_distrib2 [symmetric])
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   806
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   807
lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   808
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   809
  apply (blast intro: dvd_add)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   810
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   811
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   812
lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   813
  by (drule_tac m = m in dvd_diff, auto)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   814
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   815
lemma dvd_mult: "k dvd n ==> k dvd (m*n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   816
  unfolding dvd_def by (blast intro: mult_left_commute)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   817
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   818
lemma dvd_mult2: "k dvd m ==> k dvd (m*n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   819
  apply (subst mult_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   820
  apply (erule dvd_mult)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   821
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   822
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16796
diff changeset
   823
lemma dvd_triv_right [iff]: "k dvd (m*k :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   824
  by (rule dvd_refl [THEN dvd_mult])
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16796
diff changeset
   825
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16796
diff changeset
   826
lemma dvd_triv_left [iff]: "k dvd (k*m :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   827
  by (rule dvd_refl [THEN dvd_mult2])
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   828
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   829
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   830
  apply (rule iffI)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   831
   apply (erule_tac [2] dvd_add)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   832
   apply (rule_tac [2] dvd_refl)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   833
  apply (subgoal_tac "n = (n+k) -k")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   834
   prefer 2 apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   835
  apply (erule ssubst)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   836
  apply (erule dvd_diff)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   837
  apply (rule dvd_refl)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   838
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   839
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   840
lemma dvd_mod: "!!n::nat. [| f dvd m; f dvd n |] ==> f dvd m mod n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   841
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   842
  apply (case_tac "n = 0", auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   843
  apply (blast intro: mod_mult_distrib2 [symmetric])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   844
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   845
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   846
lemma dvd_mod_imp_dvd: "[| (k::nat) dvd m mod n;  k dvd n |] ==> k dvd m"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   847
  apply (subgoal_tac "k dvd (m div n) *n + m mod n")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   848
   apply (simp add: mod_div_equality)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   849
  apply (simp only: dvd_add dvd_mult)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   850
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   851
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   852
lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   853
  by (blast intro: dvd_mod_imp_dvd dvd_mod)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   854
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   855
lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   856
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   857
  apply (erule exE)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   858
  apply (simp add: mult_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   859
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   860
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   861
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   862
  apply auto
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   863
   apply (subgoal_tac "m*n dvd m*1")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   864
   apply (drule dvd_mult_cancel, auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   865
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   866
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   867
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   868
  apply (subst mult_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   869
  apply (erule dvd_mult_cancel1)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   870
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   871
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   872
lemma mult_dvd_mono: "[| i dvd m; j dvd n|] ==> i*j dvd (m*n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   873
  apply (unfold dvd_def, clarify)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   874
  apply (rule_tac x = "k*ka" in exI)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   875
  apply (simp add: mult_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   876
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   877
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   878
lemma dvd_mult_left: "(i*j :: nat) dvd k ==> i dvd k"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   879
  by (simp add: dvd_def mult_assoc, blast)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   880
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   881
lemma dvd_mult_right: "(i*j :: nat) dvd k ==> j dvd k"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   882
  apply (unfold dvd_def, clarify)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   883
  apply (rule_tac x = "i*k" in exI)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   884
  apply (simp add: mult_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   885
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   886
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   887
lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   888
  apply (unfold dvd_def, clarify)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   889
  apply (simp_all (no_asm_use) add: zero_less_mult_iff)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   890
  apply (erule conjE)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   891
  apply (rule le_trans)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   892
   apply (rule_tac [2] le_refl [THEN mult_le_mono])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   893
   apply (erule_tac [2] Suc_leI, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   894
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   895
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   896
lemmas dvd_eq_mod_eq_0 = dvd_def_mod [of "k\<Colon>nat" n, standard]
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   897
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   898
lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   899
  apply (subgoal_tac "m mod n = 0")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   900
   apply (simp add: mult_div_cancel)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   901
  apply (simp only: dvd_eq_mod_eq_0)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   902
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   903
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   904
lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   905
  apply (unfold dvd_def)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   906
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   907
  apply (simp add: power_add)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   908
  done
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   909
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   910
lemma mod_add_left_eq: "((a::nat) + b) mod c = (a mod c + b) mod c"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   911
  apply (rule trans [symmetric])
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   912
   apply (rule mod_add1_eq, simp)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   913
  apply (rule mod_add1_eq [symmetric])
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   914
  done
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   915
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   916
lemma mod_add_right_eq: "(a+b) mod (c::nat) = (a + (b mod c)) mod c"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   917
  apply (rule trans [symmetric])
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   918
   apply (rule mod_add1_eq, simp)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   919
  apply (rule mod_add1_eq [symmetric])
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   920
  done
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   921
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   922
lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat) | n=0)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   923
  by (induct n) auto
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   924
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   925
lemma power_le_dvd [rule_format]: "k^j dvd n --> i\<le>j --> k^i dvd (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   926
  apply (induct j)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   927
   apply (simp_all add: le_Suc_eq)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   928
  apply (blast dest!: dvd_mult_right)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   929
  done
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   930
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   931
lemma power_dvd_imp_le: "[|i^m dvd i^n;  (1::nat) < i|] ==> m \<le> n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   932
  apply (rule power_le_imp_le_exp, assumption)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   933
  apply (erule dvd_imp_le, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   934
  done
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   935
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   936
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   937
  by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16796
diff changeset
   938
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   939
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   940
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   941
(*Loses information, namely we also have r<d provided d is nonzero*)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   942
lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   943
  apply (cut_tac m = m in mod_div_equality)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   944
  apply (simp only: add_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   945
  apply (blast intro: sym)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   946
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   947
13152
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
   948
lemma split_div:
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   949
 "P(n div k :: nat) =
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   950
 ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   951
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   952
proof
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   953
  assume P: ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   954
  show ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   955
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   956
    assume "k = 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   957
    with P show ?Q by(simp add:DIVISION_BY_ZERO_DIV)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   958
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   959
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   960
    thus ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   961
    proof (simp, intro allI impI)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   962
      fix i j
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   963
      assume n: "n = k*i + j" and j: "j < k"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   964
      show "P i"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   965
      proof (cases)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   966
        assume "i = 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   967
        with n j P show "P i" by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   968
      next
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   969
        assume "i \<noteq> 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   970
        with not0 n j P show "P i" by(simp add:add_ac)
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   971
      qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   972
    qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   973
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   974
next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   975
  assume Q: ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   976
  show ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   977
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   978
    assume "k = 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   979
    with Q show ?P by(simp add:DIVISION_BY_ZERO_DIV)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   980
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   981
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   982
    with Q have R: ?R by simp
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   983
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
13517
42efec18f5b2 Added div+mod cancelling simproc
nipkow
parents: 13189
diff changeset
   984
    show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   985
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   986
qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   987
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   988
lemma split_div_lemma:
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   989
  assumes "0 < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   990
  shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m\<Colon>nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   991
proof
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   992
  assume ?rhs
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   993
  with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   994
  then have A: "n * q \<le> m" by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   995
  have "n - (m mod n) > 0" using mod_less_divisor assms by auto
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   996
  then have "m < m + (n - (m mod n))" by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   997
  then have "m < n + (m - (m mod n))" by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   998
  with nq have "m < n + n * q" by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   999
  then have B: "m < n * Suc q" by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1000
  from A B show ?lhs ..
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1001
next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1002
  assume P: ?lhs
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1003
  then have "divmod_rel m n q (m - n * q)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1004
    unfolding divmod_rel_def by (auto simp add: mult_ac)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1005
  then show ?rhs using divmod_rel by (rule divmod_rel_unique_div)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
  1006
qed
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1007
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1008
theorem split_div':
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1009
  "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1010
   (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1011
  apply (case_tac "0 < n")
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1012
  apply (simp only: add: split_div_lemma)
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1013
  apply (simp_all add: DIVISION_BY_ZERO_DIV)
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1014
  done
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1015
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1016
lemma split_mod:
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1017
 "P(n mod k :: nat) =
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1018
 ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1019
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1020
proof
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1021
  assume P: ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1022
  show ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1023
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1024
    assume "k = 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1025
    with P show ?Q by(simp add:DIVISION_BY_ZERO_MOD)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1026
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1027
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1028
    thus ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1029
    proof (simp, intro allI impI)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1030
      fix i j
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1031
      assume "n = k*i + j" "j < k"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1032
      thus "P j" using not0 P by(simp add:add_ac mult_ac)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1033
    qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1034
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1035
next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1036
  assume Q: ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1037
  show ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1038
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1039
    assume "k = 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1040
    with Q show ?P by(simp add:DIVISION_BY_ZERO_MOD)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1041
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1042
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1043
    with Q have R: ?R by simp
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1044
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
13517
42efec18f5b2 Added div+mod cancelling simproc
nipkow
parents: 13189
diff changeset
  1045
    show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1046
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1047
qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
  1048
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1049
theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1050
  apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1051
    subst [OF mod_div_equality [of _ n]])
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1052
  apply arith
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1053
  done
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
  1054
22800
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1055
lemma div_mod_equality':
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1056
  fixes m n :: nat
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1057
  shows "m div n * n = m - m mod n"
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1058
proof -
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1059
  have "m mod n \<le> m mod n" ..
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1060
  from div_mod_equality have 
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1061
    "m div n * n + m mod n - m mod n = m - m mod n" by simp
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1062
  with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1063
    "m div n * n + (m mod n - m mod n) = m - m mod n"
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1064
    by simp
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1065
  then show ?thesis by simp
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1066
qed
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1067
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1068
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
  1069
subsubsection {*An ``induction'' law for modulus arithmetic.*}
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1070
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1071
lemma mod_induct_0:
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1072
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1073
  and base: "P i" and i: "i<p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1074
  shows "P 0"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1075
proof (rule ccontr)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1076
  assume contra: "\<not>(P 0)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1077
  from i have p: "0<p" by simp
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1078
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1079
  proof
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1080
    fix k
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1081
    show "?A k"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1082
    proof (induct k)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1083
      show "?A 0" by simp  -- "by contradiction"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1084
    next
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1085
      fix n
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1086
      assume ih: "?A n"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1087
      show "?A (Suc n)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1088
      proof (clarsimp)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1089
        assume y: "P (p - Suc n)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1090
        have n: "Suc n < p"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1091
        proof (rule ccontr)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1092
          assume "\<not>(Suc n < p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1093
          hence "p - Suc n = 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1094
            by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1095
          with y contra show "False"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1096
            by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1097
        qed
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1098
        hence n2: "Suc (p - Suc n) = p-n" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1099
        from p have "p - Suc n < p" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1100
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1101
          by blast
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1102
        show "False"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1103
        proof (cases "n=0")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1104
          case True
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1105
          with z n2 contra show ?thesis by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1106
        next
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1107
          case False
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1108
          with p have "p-n < p" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1109
          with z n2 False ih show ?thesis by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1110
        qed
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1111
      qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1112
    qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1113
  qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1114
  moreover
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1115
  from i obtain k where "0<k \<and> i+k=p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1116
    by (blast dest: less_imp_add_positive)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1117
  hence "0<k \<and> i=p-k" by auto
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1118
  moreover
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1119
  note base
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1120
  ultimately
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1121
  show "False" by blast
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1122
qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1123
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1124
lemma mod_induct:
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1125
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1126
  and base: "P i" and i: "i<p" and j: "j<p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1127
  shows "P j"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1128
proof -
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1129
  have "\<forall>j<p. P j"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1130
  proof
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1131
    fix j
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1132
    show "j<p \<longrightarrow> P j" (is "?A j")
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1133
    proof (induct j)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1134
      from step base i show "?A 0"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1135
        by (auto elim: mod_induct_0)
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1136
    next
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1137
      fix k
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1138
      assume ih: "?A k"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1139
      show "?A (Suc k)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1140
      proof
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1141
        assume suc: "Suc k < p"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1142
        hence k: "k<p" by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1143
        with ih have "P k" ..
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1144
        with step k have "P (Suc k mod p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1145
          by blast
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1146
        moreover
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1147
        from suc have "Suc k mod p = Suc k"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1148
          by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1149
        ultimately
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1150
        show "P (Suc k)" by simp
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1151
      qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1152
    qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1153
  qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1154
  with j show ?thesis by blast
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1155
qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
  1156
3366
2402c6ab1561 Moving div and mod from Arith to Divides
paulson
parents:
diff changeset
  1157
end